Ebook: Dynamics in One Dimension

The behaviour under iteration of unimodal maps of an interval, such as the logistic map, has recently attracted considerable attention. It is not so widely known that a substantial theory has by now been built up for arbitrary continuous maps of an interval. The purpose of the book is to give a clear account of this subject, with complete proofs of many strong, general properties. In a number of cases these have previously been difficult of access. The analogous theory for maps of a circle is also surveyed. Although most of the results were unknown thirty years ago, the book will be intelligible to anyone who has mastered a first course in real analysis. Thus the book will be of use not only to students and researchers, but will also provide mathematicians generally with an understanding of how simple systems can exhibit chaotic behaviour.

---

The behaviour under iteration of unimodal maps of an interval, such as the logistic map, has recently attracted considerable attention. It is not so widely known that a substantial theory has by now been built up for arbitrary continuous maps of an interval. The purpose of the book is to give a clear account of this subject, with complete proofs of many strong, general properties. In a number of cases these have previously been difficult of access. The analogous theory for maps of a circle is also surveyed. Although most of the results were unknown thirty years ago, the book will be intelligible to anyone who has mastered a first course in real analysis. Thus the book will be of use not only to students and researchers, but will also provide mathematicians generally with an understanding of how simple systems can exhibit chaotic behaviour.

---

The behaviour under iteration of unimodal maps of an interval, such as the logistic map, has recently attracted considerable attention. It is not so widely known that a substantial theory has by now been built up for arbitrary continuous maps of an interval. The purpose of the book is to give a clear account of this subject, with complete proofs of many strong, general properties. In a number of cases these have previously been difficult of access. The analogous theory for maps of a circle is also surveyed. Although most of the results were unknown thirty years ago, the book will be intelligible to anyone who has mastered a first course in real analysis. Thus the book will be of use not only to students and researchers, but will also provide mathematicians generally with an understanding of how simple systems can exhibit chaotic behaviour.
Content:
Front Matter....Pages -
Introduction....Pages 1-3
Periodic orbits....Pages 5-23
Turbulence....Pages 25-46
Unstable manifolds and homoclinic points....Pages 47-67
Topological dynamics....Pages 69-89
Topological dynamics (continued)....Pages 91-119
Chaotic and non-chaotic maps....Pages 121-166
Types of periodic orbits....Pages 167-188
Topological Entropy....Pages 189-218
Maps of the circle....Pages 219-234
Back Matter....Pages -

---

The behaviour under iteration of unimodal maps of an interval, such as the logistic map, has recently attracted considerable attention. It is not so widely known that a substantial theory has by now been built up for arbitrary continuous maps of an interval. The purpose of the book is to give a clear account of this subject, with complete proofs of many strong, general properties. In a number of cases these have previously been difficult of access. The analogous theory for maps of a circle is also surveyed. Although most of the results were unknown thirty years ago, the book will be intelligible to anyone who has mastered a first course in real analysis. Thus the book will be of use not only to students and researchers, but will also provide mathematicians generally with an understanding of how simple systems can exhibit chaotic behaviour.
Content:
Front Matter....Pages -
Introduction....Pages 1-3
Periodic orbits....Pages 5-23
Turbulence....Pages 25-46
Unstable manifolds and homoclinic points....Pages 47-67
Topological dynamics....Pages 69-89
Topological dynamics (continued)....Pages 91-119
Chaotic and non-chaotic maps....Pages 121-166
Types of periodic orbits....Pages 167-188
Topological Entropy....Pages 189-218
Maps of the circle....Pages 219-234
Back Matter....Pages -
....